Small group number 4 of order 625
G is the group 625gp4
G has 2 minimal generators, rank 2 and exponent 25.
The centre has rank 2.
The 6 maximal subgroups are:
Ab(25,5) (6x).
There is one conjugacy class of maximal elementary abelian
subgroups. Each maximal elementary abelian has rank 2.
This cohomology ring calculation is complete.
Ring structure
| Completion information
| Koszul information
| Restriction information
| Poincaré series
The cohomology ring has 4 generators:
- y1 in degree 1, a nilpotent element
- y2 in degree 1, a nilpotent element
- x1 in degree 2, a regular element
- x2 in degree 2, a regular element
There are 2 minimal relations:
This minimal generating set constitutes a Gröbner
basis for the relations ideal.
Essential ideal:
There is one minimal generator:
Nilradical:
There are 2 minimal generators:
This cohomology ring was obtained from a calculation
out to degree 4. The cohomology ring approximation
is stable from degree 2 onwards, and
Carlson's tests detect stability from degree 4
onwards.
This cohomology ring has dimension 2 and depth 2.
Here is a homogeneous system of parameters:
- h1 =
x1
in degree 2
- h2 =
x2
in degree 2
The first
2 terms h1, h2 form
a regular sequence of maximum length.
The first
2 terms h1, h2 form
a complete Duflot regular sequence.
That is, their restrictions to the greatest central elementary abelian
subgroup form a regular sequence of maximal length.
The ideal of essential classes is
free of rank 1 as a module over the polynomial algebra
on h1, h2.
These free generators are:
The essential ideal squares to zero.
A basis for R/(h1, h2) is as follows.
Carlson's Koszul condition stipulates that this must
be confined to degrees less than 4.
-
1
in degree 0
-
y2
in degree 1
-
y1
in degree 1
-
y1.y2
in degree 2
Restriction to maximal subgroup number 1, which is 125gp2
- y1 restricts to
0
- y2 restricts to
y1
- x1 restricts to
x1
- x2 restricts to
x2
- y1.y2
Restriction to maximal subgroup number 2, which is 125gp2
- y1 restricts to
y1
- y2 restricts to
0
- x1 restricts to
x2
- x2 restricts to
x1
+ y1.y2
Restriction to maximal subgroup number 3, which is 125gp2
- y1 restricts to
- y1
- y2 restricts to
y1
- x1 restricts to
- x2
- x2 restricts to
x2
+ x1
- y1.y2
Restriction to maximal subgroup number 4, which is 125gp2
- y1 restricts to
2y1
- y2 restricts to
y1
- x1 restricts to
2x2
- x2 restricts to
x2
+ x1
+ 2y1.y2
Restriction to maximal subgroup number 5, which is 125gp2
- y1 restricts to
y1
- y2 restricts to
y1
- x1 restricts to
x2
- x2 restricts to
x2
+ x1
+ y1.y2
Restriction to maximal subgroup number 6, which is 125gp2
- y1 restricts to
- 2y1
- y2 restricts to
y1
- x1 restricts to
- 2x2
- x2 restricts to
x2
+ x1
- 2y1.y2
Restriction to maximal elementary abelian number 1, which is V25
- y1 restricts to
0
- y2 restricts to
0
- x1 restricts to
x2
+ x1
- x2 restricts to
- x2
Restriction to the greatest central elementary abelian, which is V25
- y1 restricts to
0
- y2 restricts to
0
- x1 restricts to
- x1
- x2 restricts to
x2
(1 + 2t + t2) /
(1 - t2)2
Back to the groups of order 625